On the oscillatory and mean motions due to waves in a thin viscoelastic layer

1. Second Jiangsu-HK Forum on Mechanics and Its Application Hohai University, Nanjing May 27 – 28, 2006 On the oscillatory and mean motions due to waves in a thin viscoelastic layer C.O. Ng, X.Y. Zhang, D.H. Zhang Department of Mechanical Engineering The University of Hong Kong Hong Kong

2. Outline • Background • Assumptions and governing equations • Results and discussions • Conclusions

3. Background • Cohesive sediments, also referred to as marine deposits or mud, are commonly found in coastal and estuarine areas. • The interaction between waves propagating on the surface of a body of water and the bed material is practically important because the waves can be attenuated at a much faster rate when the bed material is viscous and deformable than when it is rigid. • Under wave excitation, a thin layer of fluidized mud will form above a coastal bed that is made up of cohesive sediments. Depending on the concentration, the mud can behave like a viscous fluid or elastic solid or a combination of both (i.e., viscoelastic material).

4. Background • The study of water wave interacting with a viscoelastic mud has been studied by some researchers: MacPherson (1980) : the attenuation of water waves over a viscoelastic bed; while water was also assumed to be inviscid. (first order) Piedra-Cueva (1993): take into account the viscosity of water, and therefore the continuity of shear stress through the water–mud interface. (first order) Occurrence of resonance high attenuation rate Lian et al. (1999), Soltanpour et al. (2003): the mass transport velocity in a layer of viscoelastic mud under waves. (second order)

5. Background • In the above-mentioned works, the viscoelastic mud is modeled as a Voigt body: • Under simple harmonic motion: • The constitutive equation can be written as: where the complex viscoelastic parameter is whose real part is the viscosity and imaginary part is the elasticity

6. Background • Complex viscoelastic parameter: First order problem (simple harmonic) Applicable Second order problem (NOT simple harmonic) NOT Applicable

7. Objective To investigate the second-order time-mean motion/deformation undergone by a viscoelastic material when subjected to a purely oscillatory forcing, while avoiding the mistake in previous studies that the complex viscoelastic parameter is applied to the second order

8. The Present Study • Consider a thin horizontal layer of viscoelastic material of depth h subject to a periodic pressure load Ps applied on its free surface. • The material thickness h is assumed to be comparable with the boundary layer thicknessδ, but is shorter than the wave length L=2π /k:

9. Governing Equations of Motion in Lagrangian Form Continuity equation: Horizontal momentum equation: Vertical momentum equation:

10. The viscoelastic material is assumed to an isotropic Voigt body where the elastic part of the stress tensor is simply a linear function of the Lagrangian finite-strain tensor given by: Constitutive Equation Shear stress is given by:

11. Results and Discussions

12. Results and Discussions

13. Results and Discussions

14. Results and Discussions

15. Conclusions • A perturbation analysis is performed to the second order for the oscillatory and mean motions induced by a forced wave propagating in a thin viscoelastic layer. • The Lagrangian approach has been employed for the present problem. • We have avoided the mistakes made by some previous authors, who have wrongly applied the complex viscoelastic parameter to the second order.

16. Conclusions (cont’d) • We have shown that a viscoelastic material will not in general undergo a mean motion with a constant drift velocity. The mean motion will decay exponentially over a timescale equal to the ratio of the viscosity to the shear modulus. • The findings will be important to the study of Lagrangian drift over a viscoelastic bottom under surface water waves.

17. Thank youfor your attention!